Interquartile range exceeds the median What can one infer about the dispersion of a dataset when the interquartile range exceeds the median? Does this always indicate a large degree of variability? Are there any other summary statements of this dataset that can be inferred from the size of the IQR relative to the median?
 A: Note that the IQR can never be negative, but medians certainly can be negative; it's not clear that it usually makes sense to compare the two, since one is a location measure and the other is a measure of spread.
If you had data that was restricted to be always positive (no such restriction is mentioned, though), you could calculate something akin to a coefficient of variation (by calculating the ratio IQR/median)
This would be measure of relative variability, and would be unitless, like the coefficient of variation is. It might then at least make sense to ask "does such a ratio exceeding 1 indicate a large amount of relative variability?" 
However, the answer is, we can't really say; it depends on what counts as "large" for you. There's no clear absolute standard. (There isn't an absolute standard for the CV either - one that would make a particular value count as "large" or "small", though in some application areas you can find rules of thumb -- if you have some assumed distribution and some rule of thumb threshold for CV, it might be possible to find a roughly corresponding rule for IQR/median; at least perhaps under some conditions.)
A: In general, comparing the IQR to the median won't give you any extra insight about the dispersion. For example, consider these distributions:

They have the same IQR; in fact they're identical copies, just shifted along the x axis. But the IQR is greater than the median for distribution 1, and less for distribution 2. Also, consider that any distribution with median less than 0 will have IQR greater than the median.
